Theorem imaeq2 5001

Description: Equality theorem for image. (Contributed by NM, 14-Aug-1994.)

Assertion

Ref	Expression
imaeq2	|- ( A = B -> ( C " A ) = ( C " B ) )

Proof of Theorem imaeq2

Step	Hyp	Ref	Expression
1		reseq2 4937	. . 3 |- ( A = B -> ( C |` A ) = ( C |` B ) )
2	1	rneqd 4891	. 2 |- ( A = B -> ran ( C |` A ) = ran ( C |` B ) )
3		df-ima 4672	. 2 |- ( C " A ) = ran ( C |` A )
4		df-ima 4672	. 2 |- ( C " B ) = ran ( C |` B )
5	2, 3, 4	3eqtr4g 2251	1 |- ( A = B -> ( C " A ) = ( C " B ) )

Syntax hints:   -> wi 4   = wceq 1364   ran crn 4660   |` cres 4661   "cima 4662

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-xp 4665  df-cnv 4667  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672

This theorem is referenced by:  imaeq2i  5003  imaeq2d  5005  fimadmfo  5485  ssimaex  5618  ssimaexg  5619  isoselem  5863  f1opw2  6124  fopwdom  6892  ssenen  6907  fiintim  6985  fidcenumlemrk  7013  fidcenumlemr  7014  sbthlem2  7017  isbth  7026  ennnfonelemp1  12563  ennnfonelemnn0  12579  ctinfomlemom  12584  ctinfom  12585  tgcn  14376  iscnp4  14386  cnpnei  14387  cnima  14388  cnconst2  14401  cnrest2  14404  cnptoprest  14407  txcnp  14439  txcnmpt  14441  metcnp3  14679